Precalculus 3: Trigonometry

• How to solve problems in trigonometry (illustrated with 215 solved problems), in both geometrical and functional contexts, and why these methods work.
• You get a crash course in Euclidean geometry: angles, triangles, polygons, similar triangles (proportions), inscribed and circumscribed circles, bisectors, etc.
• Number pi: its definition as the ratio of the perimeter to the diameter of a disk, relation to the area of a disk, some geometrical approximations.
• The geometric definitions (by ratios in right triangles) of three trigonometric functions (sin, cos, tan) and their reciprocals (secant, cosecant, cotangent).
• Exact values of trigonometric functions for angles of 15, 18, 30, 36, 45, 60, 72, 75, and 22.5 degrees: geometric derivations, and with help of formulas.
• Solving triangles (finding side lengths and measures of all angles, knowing some of them), both right and oblique, with help of trigonometry.
• Degree vs radian: how to use proportions for recalculating degrees to radians and back; reference angles.
• The functional definition of sine, cosine and tangent, with help of unit circle and circular movement; properties of these functions.
• The definition of trigonometric (circular) functions (sin, cos, tan) for *any* real number using the unit circle in the coordinate system.
• Reference Angles Theorem with proof (by geometrical illustration) and applications; supplementary identities and the complementary angle properties.
• Periodic functions. Sinusoids: period, amplitude, phase shift, vertical shift. Transformations of graphs of trigonometric functions.
• Pythagorean theorem and Pythagorean triples. Law of Cosines, Law of Sines: formulation, proofs, and applications in problem solving.
• Various trigonometric identities with proofs, geometrical illustrations, and applications for problem solving.
• The Pythagorean Identities; Reciprocal Identities; Quotient Identities; Even/odd identities.
• Sum and Difference Identities for sine and cosine with proofs, geometrical illustrations, and applications.
• Sum To Product and Product To Sum Formulas for sine and cosine, with derivations and applications.
• Double (Half) Angles Identities with geometrical illustrations, proofs, and applications in problem solving.
• Inverse functions to sine, cosine and tangent, their definitions, properties and graphs.
• Compositions of trigonometric functions with inverse trigonometric functions; identities involving inverse trigonometric functions.
• Complex numbers and their trigonometric (polar) form; consequences of the Sum Identities for sin and cos for multiplication of complex numbers in polar form.
• De Moivre’s formula (positive natural powers of complex numbers) and its application to quick recreation of formulas for sine and cosine of multiples of angles.
• Trigonometric equations: various types and corresponding methods for solutions; depicting the solution sets on the graphs and on the unit circle.
• You get a sneak peek into trigonometry in a future Calculus class (how some trigonometric formulas are used there).
• You get a plethora of geometric illustrations, supporting your intuition and understanding of trigonometry.

Who this course is for:

• Students who plan to study Algebra, Complex Numbers, Calculus or Real Analysis
• High school students curious about university mathematics; the course is intended for purchase by adults for these students
• Everybody who wants to brush up their high school maths and gain a deeper understanding of the subject
• College and university students studying advanced courses, who want to understand all the details (concerning trigonometry) they might have missed in their earlier education
• Students wanting to learn trigonometry, for example for their College Algebra class.

Suggested Courses

• Fundamentals of Trigonometry
• Precalculus 1: Basic notions
• Linear Algebra and Geometry 1
• Become a Trigonometry & Precalculus Master
• Precalculus 2: Polynomials and rational functions